Statistical distribution of building lot depth: Theoretical and empirical investigation of downtown districts in Tokyo

Abstract

A building lot represents one of the most important basic spatial objects of urban form because building lots are basically adjacent to road networks and combine to make an exact whole with no leftover space. Thus, it is important to understand the relationship between the sizes and shapes of building lots and the density of buildings and road networks at a district scale. While building lot sizes and frontages are known to follow a log-normal distribution, respectively, the probability density function that building lot depths follow remains uncertain. Therefore, the research objective is to answer the following research question: What types of statistical distribution of building lot depths are found if the values of building density (the number of buildings per unit area) and road network density (total lengths of road networks per unit area) are given at a district scale? Assuming that (A1) one building lot has one building; (A2) building lot depth is defined as the ratio of building lot size to frontage; and (A3) building lot frontages and depths are independently distributed, I derived the probability density function of rectangular building lot depths as a log-normal distribution. As the result of theoretical investigations, it was found that (1) the probability density function of building lot depths depends not on the building density but on road network density; and (2) it depends not only on road network density but also on the variation in building lot sizes and frontages. For the empirical study of 20 downtown districts of the Tokyo metropolitan region, I tested (A3) and the log-normality of building lot depths and applied the derived function. At a 5% significance level, it was found that the hypothesis of log-normality of building lot depths was accepted in 12 of the 20 selected districts. These findings imply that when we discuss the criteria of the variation in building lot sizes and frontages, we must take into consideration the variation in building lot depths and vice versa. I also derived the probability density function of building setbacks, whose parameters include road network density and building coverage ratio. These findings are expected to provide urban planners with a theoretical basis to not only reconsider the validity of the present road network density and building coverage ratio (form-based codes) but to additionally discuss the relationship between building-lot-scale and district-scale urban physical planning.

Keywords

Building lot depth log-normal distribution density

Introduction

Buildings, building lots and road networks are significant components of urban form. Buildings plug into building lots, building lots plug into road networks and road networks all connect to form a single system in the two-dimensional plane of urban form. This set of rules is referred to as urban syntax (Marshall, 2009). Following urban syntax, urban form can be approximately regarded as a plane tessellation that basically consists of non-overlapping road networks and building lots that essentially exist adjacent to road networks (Barreira-González and Barros, 2016; Berghauser Pont and Haupt, 2009; Bitner et al., 2009; Kropf, 2018; Legras and Cavailhès, 2016; Marshall, 2009; Oliveira, 2016; Shayesteh and Steadman, 2015; Suen and Tang, 2002; Vialard and Carpenter, 2012). In urban planning, urban form has been investigated at a district scale, defined as the combination of road networks and the series of blocks (consisting of building lots and buildings) surrounded by these road networks (Berghauser Pont and Haupt, 2009; Marshall, 2009; Oliveira, 2016). In particular, a building lot represents one of the most important basic spatial objects of urban form and is called a jigsaw because building lots add up to make an exact whole with no leftover space (Marshall, 2009). Moreover, their sizes and shapes constitute the primary determinants of a residential environment and the quality of streetscapes through the size and shape of buildings on their lots (Asami, 1995a, 1995b; Asami and Niwa, 2008; Berghauser Pont and Haupt, 2010; Legras and Cavailhès, 2016; Marshall, 2009; Oliveira, 2016; Osaragi, 2014; Shayesteh and Steadman, 2015; Suen and Tang, 2002; Vialard and Carpenter, 2012). Therefore, it is important to understand the relationship between the sizes and shapes of building lots and the number of buildings and road networks at a district scale.

At a district scale, building density (the number of buildings per unit area) and road network density (total lengths of road networks per unit area) are fundamental indices in that they indicate the number of buildings and road networks in a district and provide urban planners with information regarding physical residential environments such as average building lot size in a district (Berghauser Pont and Haupt, 2010; Usui, 2018). However, based on the value of each density separately, it is difficult to understand urban form and streetscape skeletons (the basic structures of the streetscape such as the dimensions and arrangement of buildings), because urban form varies even though these values are consistent across different regions (Batty, 2009; Berghauser Pont and Haupt, 2007, 2010; Harvey and Aultman-Hall, 2016; Harvey et al., 2017). An interesting and practical method has been proposed to overcome this difficulty: the combination of several density indices, called Spacematrix, which is defined as a three-dimensional diagram whose axes are the building coverage ratio (BCR), floor area ratio (FAR) and road network density of a district. Spacematrix represents the extension of Spacemate, which is defined as a two-dimensional diagram whose axes are BCR and FAR, respectively. These methods enable us to classify urban form (Berghauser Pont and Haupt, 2004, 2007, 2010). Steadman (2014) demonstrates how further theoretical explanation can be provided for Berghauser Pont and Haupt’s empirical findings by Martin and March’s (1972) analysis. Building depth (the shorter-edge length of a rectangular building on a squared site) is introduced to facilitate a more precise morphological categorisation (Steadman, 2014). Furthermore, this rectangular building model enables us to understand the relationship between volume, wall area and plan depth (defined as the ratio of a building volume to its total exposed wall area), because real buildings can take numerous shapes whereas building depth is less easily defined (Steadman, 2006; Steadman et al., 2009). In the literature mentioned above, urban form is modelled as regular grid patterns. However, as the result of following urban syntax, building lots clearly exhibit various sizes and shapes rather than being uniform. This means that the dimension of building lots exhibits a statistical distribution. Following this, streetscape skeletons vary widely within cities. Thus, understanding this variability in terms of building density and road network density is increasingly important for developing urban design policies such as form-based codes (Harvey et al., 2017).

In the literature, building lot frontages and depths are regarded as important dimensions in the description of building lot shapes (Harvey and Aultman-Hall, 2016; Oliveira, 2016). By assuming that the shape of a building lot is rectangular, it is characterised by the combination of the frontage and depth of a building lot (Asami and Niwa, 2008; Cannaday and Colwell, 1990; Colwell and Scheu, 1989; Steadman, 2014). Whereas building lot sizes and frontages are known to follow a log-normal distribution, respectively (Ishizaka, 1984; Usui, 2018; Usui and Asami, 2019), the probability density function that building lot depths follow remains uncertain. According to Usui (2018) and Usui and Asami (2019), the parameters of the log-normal distribution can primarily be estimated by building density and road network density. However, the variation in building lot depths has yet to be investigated both theoretically and empirically. If the relationship between the statistical distribution of building lot depths and the density of buildings and road networks is also considered, we can derive more information regarding the variation in building lot shapes from the density of buildings and road networks. This meets professionals’ need for a new instrument that is able to discuss the relationship between building-lot-scale and district-scale urban physical planning. In particular, discussion of the effects of district-scale urban physical planning (building density and road network density) on the building-lot scale of urban planning is on its own insufficient.

Therefore, the research objective is to answer the following research question: What types of statistical distribution of building lot depths are found if the values of building density and road network density are given at a district scale? The answer to the research question will not only help us construct a science of cities that identifies the minimal set of mechanisms and the most important parameters to test the impact of various policies (Barthelemy, 2016), but also tackle the following practical problems. First, even if we cannot obtain data regarding building lot shapes, we can estimate the statistical distribution of building lot frontages and depths by building density and road network density. In some countries, sets of data regarding building lot shapes in a region are difficult to obtain (Gao and Asami, 2005; Wiseman and Patterson, 2016). In Japan, although cadastral maps maintained by the Land Registration Bureau represent the sole public data on land shape, they record the boundaries of land ownership rather than the boundaries of individual building lots. Furthermore, data regarding building lot shapes can be obtained from application documents for building confirmation, managed by municipal governments. However, the number of these documents is limited because they are kept for only 10 years after applications are made (Gao and Asami, 2005). Second, we can estimate the variation in building setbacks along roads. Unless building facades are legally aligned with road perimeters, the variation in building setbacks and visual road widths is dependent on the variation in building lot depths. Variability in visual road width (road width and building setback) represents one of the most important variables of the streetscape skeleton (Harvey and Aultman-Hall, 2016; Harvey et al., 2017). Until the end of the 19th century, the continuous alignment of different building facades defined the streetscape of most cities in a very clear way. However, a number of urban theories developed during the 20th century questioned this traditional alignment of building facades and instigated increasing variations in the positioning of buildings within lots by introducing zoning regulations (BCR and FAR) (Marshall, 2004, 2009, 2011; Oliveira, 2016). According to previous research, it is known that the distance of a building setback tends to be approximately half the difference of the building depth and its lot depth (Asami and Ohtaki, 2000; Usui and Asami, 2011). This means that the greater the variation in building lot depths, the greater the variation in building setbacks and visual road widths. By estimating the variation in building setbacks, the mode can also be estimated, which is a candidate of legal building setbacks under zoning regulations, in the process contributing to developing a form-based code regarding the alignment of building facades. Third, the findings concerning the relationship between the variation in building lot depths and densities of buildings and road networks are expected to contribute to exploring the potential of Spacematrix to understanding urban form not only in terms of typology but also in terms of variations in urban form at a district scale. Given that the BCR in a district can be estimated by building density (Koshizuka and Kotoh, 1989), through replacing BCR by building density, not only the variation in building lot sizes and shapes but also that of building setbacks can be positioned in the two-dimensional subspace of Spacematrix.

This paper is organised as follows. In the next section, I derive theoretically the probability density function of building lot depths with some assumptions and observe the relationship between its function shape and parameters by sensitivity analysis. Based on theoretical observations, I explore the ways in which the parameters affect the shape of the probability density function. In the third section regarding the empirical study of Tokyo downtown districts, I test the log-normality of building lot depths and apply the derived probability density function; moreover, I discuss the validity of this application and assumptions based on statistical tests. In the fourth section, I discuss the practical applications of the derived probability density function in terms of (1) how to understand the difference between the observed and theoretical distribution of building lot depths, (2) how to estimate the variation in building setbacks along roads and (3) exploring the potential of Spacematrix to understanding the variation in urban form. In the final section, I proffer concluding remarks and directions for future research.

Theoretical derivation of the probability density function of building lot depths

In this section the probability density function of building lot depths is derived theoretically with the following assumptions: (A1) one building lot has one building, (A2) building lot depth is defined as the ratio of building lot size to frontage and (A3) building lot frontages and depths are independently distributed. The rationale will be justified as follows.

First, according to the Building Standard Law of Japan (Japanese building codes), in order to appropriately regulate the size and shape of a building by BCR and FAR in its lot, a building lot basically has no more than one building, which is called the rule of one building lot for one building. This is the rationale of assuming (A1). Thus, the number of buildings is equal to that of building lots, n. This rule enables us not only to investigate the relationship between the statistical distribution of building lot depths and density indices, but also to overcome data limitations regarding building lot shapes (mentioned in the next section).

Second, although the frontage of a building lot can be uniquely defined as the intersection of a building lot and the perimeter of its front road, building lot depth cannot always be defined uniquely. Alternatively, by assuming (A2), we can overcome this difficulty. This is equivalent to assuming that the shape of a building lot is rectangular, which is analogous to the plan depth (Steadman, 2006; Steadman et al., 2009). More rigorously, for any building lot (denoted by T _i ) of a building, B _i , the area, frontage and depth of T _i are, respectively, denoted by S_i, F_i and D_i. I define F_i as the length of one side of the rectangle, which is the deformation of an arbitrary building lot shape, and D_i as the length of the other side of the rectangle, defined as

D_{i} \equiv \frac{S_{i}}{F_{i}} for \forall i

(1)

F_i and S_i deal with stochastic variables, respectively, F and S. The building lot depth, D_i, is also a stochastic variable, D, because D_i is the ratio of S_i to F_i

D \equiv \frac{S}{F}

(2)

Hereafter, S_i, F_i and D_i, respectively, denote the observed values of size, frontage and depth of T _i .

Third, according to a converse of the reproductive property, if building lot depth and frontage are two independent variables such that their product (building lot size) follows a log-normal distribution, then both building lot depth and frontage follow a log-normal distribution (Aitchison and Brown, 1957). In the literature, it has been noted that building lot sizes and frontages approximately follow a log-normal distribution, respectively (Fialkowski and Bitner, 2008; Ishizaka, 1984; Usui, 2018; Usui and Asami, 2019). The former is formulated as follows

f (S) = \frac{1}{S \sqrt{2 π σ_{S}^{2}}} e^{- \frac{{(\ln S - μ_{S})}^{2}}{2 σ_{S}^{2}}}, S > 0

(3)

In equation (3), μ_S and σ_S are, respectively, defined as the mean and standard deviation of ln S_i. This is hereafter called the law of log-normality of building lot sizes. Moreover, the probability density function of building lot frontages is formulated as follows

g (F) = \frac{1}{F \sqrt{2 π σ_{F}^{2}}} e^{- \frac{{(\ln F - μ_{F})}^{2}}{2 σ_{F}^{2}}}, F > 0

(4)

In equation (4), μ_F and σ_F are, respectively, defined as the mean and standard deviation of ln F_i. Thus, by assuming (A2) and (A3), we can derive the probability density function of building lot depths as a log-normal distribution.

Hereafter, hundreds of building lots and road networks consist of a district of area, A; the perimeter length, L; and the total road length, Λ. The unit of area and length are m² and m, respectively. The estimators of μ_S and σ_S are derived as the solutions of the simultaneous equations (E[S] = exp[μ_S + σ_S²/2], V[S] = exp[2μ_S + σ_S²](exp[σ_S²] − 1))

μ_{S} = \ln \frac{1}{\sqrt{η^{2} + 1}} \cdot \frac{κ}{ρ_{G}}

(5)

σ_{S}^{2} = \ln (η^{2} + 1)

(6)

where the estimators of E[S] and V[S] are κ/ρ_G and (ηκ/ρ_G)², respectively. In equations (5) and (6), ρ_G = n/A denotes the gross building density, η the coefficient of variation in building lot sizes and κ the ratio of the non-road area to A

κ \equiv 1 - \bar{w} {λ - \frac{1}{2} \cdot \frac{L}{A} - \bar{w} \frac{(1 + cv)}{A}} \approx 1 - \bar{w} (λ - \frac{1}{2} \cdot \frac{L}{A} - \bar{w} c \frac{v}{A}), if \frac{{\bar{w}}^{2}}{A} ≪ 1

(7)

In equation (7), L/A denotes the index reflecting the shape of a district, λ = Λ/A road network density, $\bar{w}$ mean road width, v/A the density of intersections and c the composition ratio of intersections whose degree is three and four (Usui and Asami, 2011, 2019). Moreover, the estimators of μ_F and σ_F are derived as follows

μ_{F} = \ln \frac{1}{\sqrt{η_{F}^{2} + 1}} \cdot \frac{λ (2 - \bar{w} λ)}{α ρ_{G}}

(8)

σ_{F}^{2} = \ln (η_{F}^{2} + 1)

(9)

where α denotes the ratio of the number of frontages to n and η_F denotes the coefficient of variation in building lot frontages (Usui, 2018).

Hence, assuming (A1) to (A3), if building lot sizes follow the log-normal distribution given by equation (3), building lot depths also follow a log-normal distribution

q (D) = \frac{1}{D \sqrt{2 π σ_{D}^{2}}} e^{- \frac{{(\ln D - μ_{D})}^{2}}{2 σ_{D}^{2}}}, D > 0

(10)

In equation (10), μ_D and σ_D are, respectively, defined as the mean and standard deviation of ln D_i. The estimators of μ_D and σ_D are derived as follows

μ_{D} = μ_{S} - μ_{F} = \ln \frac{\sqrt{η_{F}^{2} + 1}}{\sqrt{η^{2} + 1}} \cdot \frac{α κ}{λ (2 - \bar{w} λ)}

(11)

\begin{array}{l} σ_{D}^{2} = σ_{S}^{2} - σ_{F}^{2} - 2 Cov (\ln F, \ln D) = \ln \frac{η^{2} + 1}{η_{F}^{2} + 1}, where η_{F} < η \end{array}

(12)

In equation (11), μ_D = E[ln(S/F)] = E[lnS] – E[lnF] = μ_S – μ_F. Also, since (A3) is assumed, Cov(F, D) equals zero. This is equivalent to Cov(ln F, ln D) = 0. Therefore, substituting equations (11) and (12) into equation (10), the probability density function of building lot depths is derived as

q (D) = \frac{1}{D \sqrt{2 π \ln \frac{η^{2} + 1}{η_{F}^{2} + 1}}} e^{- \frac{{\ln D - \ln \frac{\sqrt{η_{F}^{2} + 1}}{\sqrt{η^{2} + 1}} \cdot \frac{α κ}{λ (2 - \bar{w} λ)}}^{2}}{2 \ln \frac{η^{2} + 1}{η_{F}^{2} + 1}}}, D > 0, η_{F} < η

(13)

From equation (13), it is found that (1) q(D) does not depend on ρ_G and (2) q(D) is defined if the condition η_F < η is satisfied. Therefore, it is α rather than ρ_G that affects the shape of q(D). As Oliveira (2016) suggests, the plot system (the set of building lots) has less durability than road networks. This implies that the shape of q(D) primarily depends on λ. Thus, ρ_G represents a secondary parameter following λ, on which the variation in building lot sizes and shapes depends. Moreover, awareness of the above condition may be useful to urban planners because the variation in building lot frontages is restricted by the variation in building lot sizes.

From these theoretical observations (see also online supplemental material Figure S1), I suggest that the shape of q(D) depends on not only λ but also the difference between η and η_F. Therefore, in actual urban space, knowledge of the degree of variation in the values of η and η_F is important in estimating the statistical distribution of building lot depths. If the values of η and η_F tend to be constant regardless of differences between districts, then the statistical distribution of building lot depths can be estimated from only λ considering stability of the other parameters. Hereafter, I call this the hypothesis of stability of η and η_F and confirm whether the hypothesis holds or not in the next section. Even if this hypothesis does not hold, the results of the sensitivity analysis from Figure S1 provide useful information regarding the effect of changing the value of each parameter on the shape of q(D). In particular, urban planners may be interested to learn that the variation in building lot depths depends on the variation in building lot sizes and frontages. This suggests that when determining the criteria behind the variation in building lot sizes and frontages we must consider the variation in building lot depths and vice versa.

Empirical study of downtown districts in Tokyo

In the previous section, in theory q(D) can be derived as a log-normal distribution with three assumptions. In practice, however, whether or not building lot depths follow a log-normal distribution has not been confirmed. In addition, whether (A3) holds or not remains unconfirmed. Thus, their validity will be confirmed based on empirical data and statistical tests.

Empirical study area

This empirical study of an actual urban area is focused on a downtown ward of the Tokyo metropolitan eastern region, Sumida Ward. Since the second half of the 20th century, the Tokyo metropolitan region has experienced a steady diminution in building lot sizes with the continuing subdivision of building lots in order to meet significant demand for housing (Osaragi, 2014). As a result of this process, building lots with small sizes, narrow frontages and great depths are generated with insufficient road network density, especially in the northern districts of Sumida Ward (Usui and Asami, 2019). In order to improve the residential environment, redevelopment projects are ongoing in Tokyo downtown districts, and may drastically change building lot sizes and shapes by 2020, when Tokyo will host the Olympic Games. As Figure 1 (left) shows, whereas the northern districts of Sumida Ward display irregular patterns of building lots as a result of unplanned urbanisation, the southern districts demonstrate a regular pattern as a result of land readjustment projects implemented after Second World War. The building lot patterns of the former and latter contrast with one another. The building types and building stories in Sumida ward are shown in online supplemental material Figure S2. Among all buildings, 45% and 12% are detached housing and housing complexes, respectively. In the following, 31% and 5% of all buildings are commercial with housing or factories with housing and offices or commercial buildings, respectively. As for building stories, approximately 60% of all buildings have one to two stories. Buildings with more than six stories tend to be distributed along wide road networks (road width is wider than 10 metres) as the result of legal zoning regulations (FAR along wide roads is higher than narrow ones).

Figure 1.

The patterns of building lots of Sumida Ward (left) and the districts selected for empirical analysis (right).

In order to analyse the relationship between the statistical distribution of building lot depths and the density of buildings and roads, I selected 20 districts from the northern and southern region of Sumida Ward based on the combination of ρ_G and λ, as shown in Table S1 (see online supplemental material). The locations of these districts are illustrated in Figure 1 (right). In Tokyo’s 23 wards, the typical combination of ρ_G and λ is 0.003 and 0.03, respectively. In Sumida Ward, their values are 0.004 and 0.04, respectively.

Data

As mentioned above, data regarding building lot shapes are difficult to obtain in Japan. Gao and Asami (2005) have proposed a method to estimate the boundary lines of a building lot, as opposed to the boundary lines of building lots in a district. However, the latter method has remained underdeveloped. In response, Hamaina et al. (2014) have proposed that data regarding the shape of area Voronoi cells – whose generators are building polygons – overcome not only the potential lack of data availability regarding building lot shapes, but also potential spatial coverage problems between buildings and plot layers. In this paper, to substitute for the set of data regarding the shape of actual building lots in a district, I focus on the set of data on the shape of area Voronoi cells {V(B _i )}, whose generators are building polygons, {B _i }, and the centre line of road networks, R, in a district (Usui and Asami, 2010, 2013). An area Voronoi diagram represents an extension of an ordinary Voronoi diagram, whose generators are the set of points (Okabe et al., 2000). The generators of an area Voronoi diagram are the set of polylines or polygons that, respectively, correspond to the centre line of road networks, R, and the edges of building polygons. As substitutes for the set of the shape of actual building lots, I use the intersection of {V(B _i )} and urban blocks, denoted by T _i (see online supplemental material Figure S3). The frontage of T _i , denoted by F_i, is defined as the intersection of T _i and the perimeter of the urban block on which B _i is located. The details of the spatial data used are shown in Table 1. This substitution corresponds to (A1) in the second section. Thus, (A1) enables us not only to investigate the relationship between ρ_G and the variation in building lot shapes but also to overcome the lack of data regarding actual building lot shapes. On the other hand, a limitation of these substitutes is that the frontage and size of T _i may be either larger or smaller than those of its actual building lot. Indeed, accuracy is contingent on the difference in setback distance along roads among B _i and both left- and right-side buildings. Due to a lack of data availability, accuracy needs to be confirmed in future research.

Table 1.

Details of spatial data used for empirical analysis.

Spatial data	Source
Districts	Statistical GIS provided by Statistics bureau, Ministry of Internal Affairs and Communications, Japan.
Buildings	Residential Maps by Zenrin, Co., Ltd
Centreline of roads and urban blocks	Mapple 10000 Digital Data released by Shobunsha Publications, Inc.

Test for the log-normality of building lot depths

In order to statistically test the log-normality of observed building lot depths, ln D_i, I adopt a Kolmogorov–Smirnov test (KS test), whose test statistic is defined as

H_{n} \equiv \max_{\ln D_{j}} {| F (\ln D_{j}) - G (\ln D_{j}) |}

(14)

where D_j denotes the ascending order of D_i, (j = 1, …, n), F(ln D_j) the theoretical cumulative probability of ln D_j (e.g. log-normal distribution) and G(ln D_j) the observed cumulative probability of ln D_j. The critical values of the KS test at a 5% significance level and at a 1% significance level are

H_{0} (n) \equiv 1.36 / \sqrt{n}

and

H_{0} (n) \equiv 1.63 / \sqrt{n}

, respectively.

In Table 2 (the second column), I show the results of the KS tests for the log-normality of building lot depths. It is found that (1) at a 5% significance level, the hypothesis of log-normality of building lot depths is accepted in 12 of the 20 selected districts; and (2) at a 1% significance level, the hypothesis is accepted in the selected districts, with the exception of the following five districts: Higashi-Mukōjima 3, Kyōjima 1, Higashi-Mukōjima 4, Kyōjima 3 and Sumida 3. In Figure 2, the observed distribution of building lot depths in 6 of the 20 selected districts is represented by the grey-coloured relative frequency distribution. The log-normal distribution given by equation (13) is represented by the heavy solid line. In the 20 districts selected, it was found that the values of η_F and η tend to be stable in the neighbourhood of 0.60 and 1.02, respectively (Usui, 2018; Usui and Asami, 2019). Thus, it can be concluded that the hypothesis of stability of η and η_F tends to hold in empirical study areas.

Table 2.

Result of the KS tests for log-normality of building lot depths.

Districts	H_n	H₀(n)
Sumida 1	0.080*	0.073 (343)
Kamezawa 2	0.069	0.078 (301)
Oshiage 1	0.055*	0.047 (821)
Tatekawa 4	0.055	0.077 (308)
Adumabashi 2	0.045	0.079 (294)
Kinshi 2	0.061	0.092 (218)
Higashi-Mukōjima 3	0.075**	0.047 (843)
Tatekawa 3	0.078	0.078 (301)
Mukōjima 3	0.042	0.044 (959)
Honjo 4	0.050	0.059 (534)
Taihei 3	0.046	0.080 (289)
Kyōjima 1	0.223**	0.041 (1075)
Higashi-Mukōjima 4	0.092**	0.041 (1108)
Tatekawa 1	0.042	0.073 (343)
Ishihara 3	0.038	0.060 (514)
Yahiro 4	0.043*	0.040 (1132)
Higashi-Mukōjima 5	0.041	0.041 (1078)
Honjo 3	0.049	0.058 (546)
Kyōjima 3	0.099**	0.032 (1776)
Sumida 3	0.055**	0.037 (1316)

KS: Kolmogorov–Smirnov.

H₀(n) denotes the critical value of KS test at a 5% significance level.

* indicates significant at a 5% level. ** indicates significant at a 1% level.

Figure 2.

The observed and theoretical distributions of building lot depths. (a) Tatekawa 4, (b) Mukōjima 3, (c) Tatekawa 1, (d) Kinshi 2, (e) Honjo 3 and (f) Higashi-Mukōjima 3**.

Moreover, through sensitivity analysis it was found that the change in the value of L/A, v/A and c in equations (7) and (13) scarcely affects the shape of q(D). Thus, I conclude that (1) the shape of a district, the number of intersections and the composition ratio of intersections of order 3 and order 4 barely affect the statistical distribution of building lot depths; and (2) the statistical distribution of building lot depths can be estimated from only λ considering stability of the other parameters.

Test for independence of building lot frontages and depths

In order to statistically test the hypothesis that building lot frontages and depths are independently distributed ((A3) in the second section), I adopt the following non-parametric test statistic (Usui and Asami, 2013)

Z = \frac{\bar{s} - \bar{F} \bar{D}}{\sqrt{V_{un} [\bar{s} (t_{k})]}} \sim N (0, 1), V_{un} [\bar{s} (t_{k})] = \frac{1}{m} \cdot \frac{\sum_{i = 1}^{m} \sum_{j = 1}^{m} {(F_{i} D_{j} - \bar{F} \bar{D})}^{2}}{m^{2} - 1}

(15)

In equation (15), $\bar{s}$ , $\bar{F}$ and $\bar{D}$ , respectively, denote the mean of the building lot size, frontage and depth. Moreover, m denotes the number of pairs of building lot frontage and depth. If this hypothesis holds, at a 5% significance level the value of |Z| < 1.96. It was found that this hypothesis is accepted in the selected districts except for Sumida 1, Kinshi 2 and Sumida 3. Hence, it is empirically confirmed that (A3) tends to be satisfied. Thus, we can theoretically derive the probability distribution function of building lot depths as the log-normal distribution given by equation (13).

Discussion

Based on these theoretical and empirical findings, I discuss the practical applications and limitation of the derived probability density function in terms of (1) how to understand the difference between the observed and theoretical distribution of building lot depths, (2) how to estimate the variation in building setbacks along roads and (3) exploring the potential of Spacematrix to understand the variation in urban form.

First, according to the sensitivity analyses presented in Figure S1, λ is specified as most sensitive to the shape of q(D), given by equation (13) among its parameters. However, the irregularity of building lot shapes cannot be taken into account due to (A2). According to Osaragi (2014), the following two basic subdivision patterns can be typically observed in districts in Tokyo: simple division and flag-shaped division (see online supplemental material Figure S4). Simple division is the subdivision of a regular building lot into two regular ones. On the other hand, flag-shaped division is often seen in built-up districts, resulting from extensive subdivisions of a regular building lot into regular and flag-shaped ones in order to be adjacent to their front road. If the shape of a building lot is flag shaped, its depth per se and the ratio of its depth to frontage are overestimated because its frontage is narrower than the rectangular building lot. According to Gao and Asami (2007), the frontages of flag-shaped building lots are usually 2.0 to 3.0 metres, while for regular building lots the frontages are at least approximately 4.0 metres (3.0 metres for a room plus necessary space for structures and distance to neighbours). Therefore, from the value of the frontage, we can roughly distinguish regular lots from flag-shaped ones. Unfortunately, given that spatial data regarding whether a building lot is flag shaped or not cannot be obtained in Japan, we compare the relative number of flag-shaped building lots among the districts selected by investigating the relationship between the estimated ratio of building lot depth to frontage based on an allometric approach and the average building lot frontage, which can be approximately estimated by $\bar{F} \approx λ (2 - \bar{w} λ) / α ρ_{G} \approx 2 λ / ρ_{G}$ (equivalent to 2Λ/n) (Usui, 2018).

In order to understand what happens to building lot shapes as the density of buildings and road networks changes at a district scale, I estimate the ratio of building lot depth to frontage by investigating the allometric relationship between building lot sizes, frontages and depths

\hat{F_{i}} = S_{i} {\hat{β}}_{F}

(16)

\hat{D_{i}} = S_{i} {\hat{β}}_{D}

(17)

where

{\hat{β}}_{F}

and

{\hat{β}}_{D}

denote the estimated allometric coefficient regarding building lot frontage and depth, respectively. It was found that, in the selected districts, (1)

{\hat{β}}_{F}

ranges from 0.455 (Kyōjima 3) to 0.477 (Taihei 3) (R² is more than 0.943); and (2)

{\hat{β}}_{D}

ranges from 0.522 (Taihei 3) to 0.544 (Kyōjima 3) (R² is more than 0.959). In this case, the ratio of building lot depth to frontage can be estimated:

{\hat{D}}_{i} / {\hat{F}}_{i} = S_{i} {\hat{β}}_{D} - {\hat{β}}_{F}

{\hat{β}}_{D} - {\hat{β}}_{F}

ranges from 0.05 (Taihei 3 and Kamezawa 3) to 0.09 (Kyōjima 1, Higashi-Mukōjima 4 and Kyōjima 3). The relationship between 2λ/ρ_G and

{\hat{D}}_{i} / {\hat{F}}_{i}

in the selected districts is displayed in Figure S5 (see online supplemental material). It was found that the correlation coefficient between 2λ/ρ_G and

{\hat{D}}_{i} / {\hat{F}}_{i}

-

0.76, which indicates that

{\hat{D}}_{i} / {\hat{F}}_{i}

is negatively correlated with 2λ/ρ_G. Hence, where 2λ/ρ_G is smaller, the building lot frontage and depth are shorter and relatively longer, respectively. In the selected districts, 2λ/ρ_G ranges from 9.7 metres (Kyōjima 3) to 26.3 metres (Kamezawa 2). Also, it was confirmed that 2λ/ρ_G and

{\hat{D}}_{i} / {\hat{F}}_{i}

in these 8 districts tend to be smaller and larger than those in the other 12 districts. If simple division were more pronounced than flag-shaped division in these eight districts,

{\hat{D}}_{i} / {\hat{F}}_{i}

could be smaller than the present value. This implies that the number of flag-shaped building lots in these 8 districts is relatively larger than that in the other 12 districts. Thus, it should be appreciated that the more pronounced the flag-shaped building lots, the less the observed distribution of building lot depths tends to be positively skewed compared with the theoretical distribution of building lot depths. This is a limitation of the applicability of the theoretical distribution given by equation (13). On the other hand, the reduction of the difference may be regarded as the reduction of flag-shaped building lots to regular ones in a district.

Second, we can derive the probability density function of building setbacks. Assuming that (1) the shapes of a building and its lot are similar to each other and are rectangular; and (2) the gravity locations of a building and its lot are identical to each other, the distance of a building setback can be derived as half of the difference of the building depth and its lot depth (Asami and Ohtaki, 2000; Usui and Asami, 2011)

U = \frac{1}{2} (1 - \sqrt{τ}) D

(18)

where τ denotes the BCR in its lot, which is legally given in a district. From equations (13) and (18), the probability density function of building setbacks can be derived as

z (U) = \frac{1 - \sqrt{τ}}{2 U \sqrt{2 π \ln \frac{η^{2} + 1}{η_{F}^{2} + 1}}} e^{- \frac{{\ln U - \ln \frac{1 - \sqrt{τ}}{2} - \ln \frac{\sqrt{η_{F}^{2} + 1}}{\sqrt{η^{2} + 1}} \cdot \frac{α κ}{λ (2 - λ \bar{w})}}^{2}}{2 \ln \frac{η^{2} + 1}{η_{F}^{2} + 1}}}, U > 0, η_{F} < η

(19)

Figure 3 shows the probability density functions of D and U, respectively. It was found that (1) if λ = 0.03, the mode of setbacks is 3 metres; (2) if λ = 0.04, the mode of setbacks is 2 metres and (3) the maximum setback distance is 15 metres. According to Japanese building codes, setback distance along roads in low rise and exclusive residential districts must be either 1 or 1.5 metres. These findings help provide a rationale for the deterministic setback distance because the typical setback distance can be estimated by λ and τ. Moreover, the shape of z(U) (λ = 0.03, τ = 0.7) is approximately the same as that of z(U) (λ = 0.04, τ = 0.6). This means that λ (district-scale urban planning) and τ (building-lot-scale urban planning) are equivalent to one another. Thus, the variation in building setbacks can be controlled by altering either λ or τ. These findings are expected to provide urban planners with a theoretical basis to reconsider the validity of the present λ and τ in order to improve the continuous alignment of different buildings under zoning regulation. They additionally offer a rationale for determining the visual road width of the streetscape skeleton (Harvey et al., 2017). They set the visual road width of the streetscape skeleton at 40 metres, which is approximately equal to the average road width (10 metres) and twice as long as the maximum setback distance (30 metres).

Figure 3.

The probability density functions of D and U, respectively (τ = 0.6). (a) λ = 0.03 and (b) λ = 0.04.

Third, I show that Spacematrix has the potential to provide information not only regarding the typology of urban form but also concerning the variation in building lot sizes and shapes and building setbacks through the application of the aforementioned findings. The shapes of f(S) and g(F) depend on both ρ_G and λ (Usui, 2018; Usui and Asami, 2019). As mentioned in the second section, however, the shape of q(D) depends not on ρ_G but on λ. Thus, in terms of the variation in building lot sizes, frontages and depths, λ should be regarded as the primary axis of Spacematrix. This finding strengthens the suggestion that adding λ as a primary indicator of the density concept improves its capacity to indicate the important primary measurements of the urban landscape and describe important aspects of urban form (Berghauser Pont and Haupt, 2010). Spacematrix is shown in online supplemental material Figure S6. Considering that parameters other than λ and ρ_G scarcely affect the shapes of f(S), g(F) and q(D) (Usui, 2018; Usui and Asami, 2019), once the present values of λ and ρ_G are computed, the present variation in building lot sizes and shapes at a district scale can easily be estimated and characterised in the two-dimensional subspace of Spacematrix. Furthermore, legally given τ in a district, the present variation in building setbacks at a district scale can be characterised in the two-dimensional subspace of Spacematrix. Moreover, the application of the findings to Spacematrix provides a theoretical framework to reconsider the validity of the present values of λ, ρ_G and τ, termed form-based codes, in order to promote a residential environment and well-arranged streetscape under zoning regulation.

Conclusions

The research objective was to answer the following research question: What types of statistical distribution of building lot depths are found if the values of building density and road network density are given at a district scale?

Assuming (A1) to (A3), I theoretically derived the probability density function of building lot depths as a log-normal distribution, q(D). As the result of theoretical investigations, it was found that (1) q(D) depends not on ρ_G but on α; and (2) q(D) is defined if the condition of η_F < η is satisfied. The former implies that the shape of q(D) primarily depends on λ. The latter implies that the variation in building lot frontages is restricted by that of building lot sizes. Although the shapes of f(S) and g(F) are contingent on both ρ_G and λ, the shape of q(D) depends on λ and α. In the literature, α can be regarded as constant. Thus, the following two steps (albeit not simultaneously) should be adopted: first, the criteria of λ should be determined based on the relationship between q(D) and λ; second, for given λ, the criteria of ρ_G should be determined based on the relationship between f(S) and g(F). In the literature, these properties have not been known as an urban physical planning theory, yet they are of critical importance in terms of the relationship between urban form and densities of buildings and road networks. For the empirical study, I tested the hypothesis that building lot frontages and depths are independently distributed (A3), tested the log-normality of building lot depths and applied the derived probability density function to 20 downtown districts of the Tokyo metropolitan region, discussing the validity of this application based on statistical tests. It was found that (1) at a 5% significance level, building lot frontages and depths tend to be independently distributed based on a non-parametric test; and (2) at a 1% significance level, building lot depths generally follow a log-normal distribution in the empirical study districts, with the exception of five districts where 2λ/ρ_G is smaller than that in the other 15 districts. These findings constitute the answer to the research questions. As discussed in the previous section, these findings are expected to provide urban planners with a theoretical basis to reconsider the validity of the present values of λ, ρ_G and τ, termed form-based codes, in order to improve the residential environment and well-arranged continuous alignment of different buildings.

Several points are suggested for future research. First, although assumption (A1), the rule of one building lot for one building, enables us not only to investigate the relationship between ρ_G and the variation in building lot shapes but also to overcome the lack of data regarding building lot shapes, there are many forms of development where this rule does not apply. In this case, assumption (A1) is a significant limitation on the wider applicability of the findings. Thus, if data regarding actual building lot shapes are obtainable in the future, the validity of assumption (A1), the accuracy of the approximation of an actual building lot shape to a rectangular one and their effects on the estimation of building lot depths need to be confirmed in detail. Second, the theoretical distribution of building lot depths must be applied to additional urban areas in other countries, especially in Western and Asian countries, in order to confirm whether or not the log-normality of building lot depths and the hypothesis of stability of η_F and η hold. Third, it would be possible to estimate the statistical distribution of building lot depths not only in the past but also in the future using the density of buildings and road networks. Finally, the analysis can be considered as the first step in the analysis of the streetscape skeleton by focusing on building lot depths. In future analysis it will be important to investigate not only the variation in building lot depths but also the variation in building lot frontages, front road width and building heights in order to advance form-based codes that take these variations into consideration.

Supplemental Material

Supplemental material for Statistical distribution of building lot depth: Theoretical and empirical investigation of downtown districts in Tokyo

Supplemental material for Statistical distribution of building lot depth: Theoretical and empirical investigation of downtown districts in Tokyo by Hiroyuki Usui in EPB: Urban Analytics and City Science

Footnotes

Acknowledgements

The author is grateful to Professor Yasushi Asami and two anonymous referees for their extremely valuable comments and suggestions. This research was the result of the joint research with CSIS, the University of Tokyo (No. 785) and used the following data: Residential Maps provided by Zenrin, CO., Ltd.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received the following financial support for the research, authorship, and/or publication of this article: This work was supported by Japan Society for the Promotion of Science (16H01830 and 17K12978).

Supplemental material

Supplemental material for this article is available online.

Hiroyuki Usui received PhD in Engineering from the University of Tokyo in 2014 and now is an assistant professor of Department of Urban Engineering, the University of Tokyo. His research interests are city planning, spatial analysis, urban mathematical models and urban form. He studies the relationship of urban density and urban form. He published several papers in spatial analysis related journals, such as International Journal of Geographical Information Science, International Regional Science Review, Journal of Geographical Systems and Cybergeo: European Journal of Geography. He teaches urban planning and geographical information systems to undergraduate and graduate students.

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